Using Part 2 of the Fundamental Theorem of Calculus to find the derivative of an integral Hey I've been really struggling with the following problem any help would be greatly appreciated. My rep is currently too low so I have to paste the image sorry for any inconvenience! I need to find the derivative of this equation $$F(x)=\int_{5x}^{2}\sin(t^3)\,dt$$

$$F (x)=\int_{5x}^2\sin (t^3)dt $$

put

$$G (X)=\int_0^X \sin (t^3)dt. $$

$t\mapsto \sin (t^3) $ is continuous at $\mathbb R$, thus $G $ is differentiable at $\mathbb R $ and By FTC, $$G'(x)=\sin (x^3) . $$

but

$$F (x)=G (2)-G (5x) $$

then by chain rule, $F $ is differentiable at $\mathbb R $ and

$$(\forall x\in \mathbb R)\;\; \; F'(x)=-(G'(5x ))\times 5$$

> $$=-5\sin (125x^3) $$